# Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

# Linear Systems: Continuous-Time Impulse Response Descriptions

Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_188-1

## Abstract

An important input–output description of a linear continuous-time system is its impulse response, which is the response h(t, τ) to an impulse applied at time τ. In time-invariant systems that are also causal and at rest at time zero, the impulse response is h(t, 0) and its Laplace transform is the transfer function of the system. Expressions for h(t, τ) when the system is described by state-variable equations are also derived.

## Keywords

Linear systems Continuous-time Time-invariant Time-varying Impulse response descriptions Transfer function descriptions

## Introduction

Consider linear continuous-time dynamical systems, the input–output behavior of which can be described by an integral representation of the form
$$y(t) =\displaystyle\int _{ -\infty }^{+\infty }H(t,\tau )u(\tau )d\tau$$
(1)
where $$t,\tau \in \mathbb{R}$$, the output is $$y(t) \in {\mathbb{R}}^{p}$$, the input is $$u(t) \in {\mathbb{R}}^{m}$$, and $$H : \mathbb{R} \times \mathbb{R} \rightarrow {\mathbb{R}}^{p\times m}$$ is assumed to be integrable. For instance, any system in state-variable form
$$\begin{array}{l} \dot{x} = A(t)x + B(t)u \\ y = C(t)x + D(t)u\\ \end{array}$$
(2)
or
$$\begin{array}{l} \dot{x} = Ax + Bu \\ y = Cx + Du\\ \end{array}$$
(3)
also has a representation of the form (1) as we shall see below.
Note that it is assumed that at $$\tau = -\infty$$, the system is at rest. H(t, τ) is the impulse response matrix of the system (1). To explain, consider first a single-input single-output system:
$$y(t) =\displaystyle\int _{ -\infty }^{+\infty }h(t,\tau )u(\tau )d\tau ,$$
(4)
and recall that if $$\delta (\hat{t}-\tau )$$ denotes an impulse (delta or Dirac) function applied at time $$\tau =\hat{ t}$$, then for a function f(t),
$$f(\hat{t}) =\displaystyle\int _{ -\infty }^{+\infty }f(\tau )\delta (\hat{t}-\tau )d\tau .$$
(5)
If now in (4) $$u(\tau ) =\delta (\hat{t}-\tau )$$, that is, an impulse input is applied at $$\tau =\hat{ t}$$, then the output y I (t) is
$$y_{I}(t) = h(t,\hat{t}),$$
i.e., $$h(t,\hat{t})$$ is the output at time t when an impulse is applied at the input at time $$\hat{t}$$. So in (4), h(t, τ) is the response at time t to an impulse applied at time τ. Clearly if the impulse responseh(t, τ) is known, the response to any input u(t) can be derived via (4), and so h(t, τ) is an input/output description of the system.
Equation (1) is a generalization of (4) for the multi-input, multi-output case. If we let all the components of u(τ) in (1) be zero except the jth component, then
$$y_{i}(t) =\displaystyle\int _{ -\infty }^{+\infty }h_{ ij}(t,\tau )u_{j}(\tau )d\tau ,$$
(6)
h ij (t, τ) denotes the response of the ith component of the output of system (1) at time t due to an impulse applied to the jth component of the input at time τ with all remaining components of the input being zero. H(t, τ) = [h ij (t, τ)] is called the impulse response matrix of the system.
If it is known that system (1) is causal, then the output will be zero before an input is applied. Therefore,
$$H(t,\tau ) = 0,\quad \mathrm{for}\ t <\tau ,$$
(7)
and (1) becomes
$$y(t) =\displaystyle\int _{ -\infty }^{t}H(t,\tau )u(\tau )d\tau .$$
(8)
Rewrite (8) as
$$\displaystyle\begin{array}{rcl} y(t)& =& \displaystyle\int _{-\infty }^{t_{0} }H(t,\tau )u(\tau )d\tau +\displaystyle\int _{ t_{0}}^{t}H(t,\tau )u(\tau )d\tau \\ & =& y(t_{0}) +\displaystyle\int _{ t_{0}}^{t}H(t,\tau )u(\tau )d\tau . \end{array}$$
(9)
If (1) is at rest at t = t 0 (i.e., if u(t) = 0 for tt 0, then y(t) = 0 for tt 0), y(t 0) = 0 and (9) becomes
$$\begin{array}{rlrlrl} y(t) & =\displaystyle\int _{ t_{0}}^{t}H(t,\tau )u(\tau )d\tau . &\end{array}$$
(10)
If in addition system (1) is time-invariant, then $$H(t,\tau ) = H(t-\tau ,0)$$ (also written as H(tτ)) since only the elapsed time (tτ) from the application of the impulse is important. Then (10) becomes
$$y(t) =\displaystyle\int _{ 0}^{t}H(t-\tau )u(\tau )d\tau ,\quad t \geq 0,$$
(11)
where we chose t 0 = 0 without loss of generality. Equation (11) is the description for causal, time-invariant systems, at rest at t = 0.
Equation (11) is a convolution integral and if we take the (one-sided or unilateral) Laplace transform of both sides,
$$\hat{y}(s) =\hat{ H}(s)\hat{u}(s),$$
(12)
where $$\hat{y}(s),\ \hat{u}(s)$$ are the Laplace transforms of y(t), u(t) and $$\hat{H}(s)$$ is the Laplace transform of the impulse response H(t). $$\hat{H}(s)$$ is the transfer function matrix of the system. Note that the transfer function of a linear, time-invariant system is typically defined as the rational matrix $$\hat{H}(s)$$ that satisfies (12) for any input and its corresponding output assuming zero initial conditions, which is of course consistent with the above analysis.

## Connection to State-Variable Descriptions

When a system is described by the state-variable description (2), then
$$y(t) =\displaystyle\int _{ t_{0}}^{t}[C(t)\Phi (t,\tau )B(\tau ) + D(t)\delta (t-\tau )]u(\tau )d\tau ,$$
(13)
where it was assumed that x(t 0) = 0, i.e., the system is at rest at t 0. Here $$\Phi (t,\tau )$$ is the state transition matrix of the system defined by the Peano-Baker series:
$$\Phi (t,t_{0}) = I +\displaystyle\int \limits _{ t_{0}}^{t}A(\tau _{ 1})d\tau _{1}+\displaystyle\int \limits _{t_{0}}^{t}A(\tau _{ 1})\left [\displaystyle\int \limits _{t_{0}}^{\tau _{1} }A(\tau _{2})d\tau _{2}\right ]d\tau _{1} + \cdots \,;$$
see “Linear Systems: Continuous-Time, Time-Varying, State Variable Descriptions,” Panos J. Antsaklis.
Comparing (13) with (10), the impulse response
$$H(t,\tau ) = \left \{\begin{array}{@{}l@{\quad }l@{}} C(t)\Phi (t,\tau )B(t) + D(t)\delta (t-\tau )\quad &t \geq \tau ,\\ 0 \quad &t <\tau . \end{array} \right .$$
(14)
Similarly, when the system is time-invariant and is described by (3),
$$y(t) =\displaystyle\int _{ t_{0}}^{t}[C{e}^{A(t-\tau )}B + D\delta (t-\tau )]u(\tau )d\tau ,$$
(15)
where x(t 0) = 0.
Comparing (15) with (11), the impulse response
$$H(t-\tau ) = \left \{\begin{array}{@{}l@{\quad }l@{}} C{e}^{A(t-\tau )}B + D\delta (t-\tau )\quad &t \geq \tau , \\ 0 \quad &t < 0, \end{array} \right .$$
(16)
or as it is commonly written (taking the time when the impulse is applied to be zero, τ = 0)
$$H(t) = \left \{\begin{array}{@{}l@{\quad }l@{}} C{e}^{At}B + D\delta (t)\quad &t \geq 0, \\ 0 \quad &t < 0. \end{array} \right .$$
(17)
Take now the (one-sided or unilateral) Laplace transform of both sides in (17) to obtain
$$\hat{H}(s) = C{(sI - A)}^{-1}B + D,$$
(18)
which is the transfer function matrix in terms of the coefficient matrices in the state-variable description (3). Note that (18) can also be derived directly from (3) by assuming zero initial conditions (x(0) = 0) and taking Laplace transform of both sides.

Finally, it is easy to show that equivalent state-variable descriptions give rise to the same impulse responses.

## Summary

The continuous-time impulse response is an external, input–output description of linear, continuous-time systems. When the system is time-invariant, the Laplace transform of the impulse response h(t, 0) (which is the output response at time t due to an impulse applied at time zero with initial conditions taken to be zero) is the transfer function of the system – another very common input–output description. The relationships with the state-variable descriptions are shown.

## Recommended Reading

External or input–output descriptions such as the impulse response and the transfer function (in the time-invariant case) are described in several textbooks below.

## Bibliography

1. Antsaklis PJ, Michel AN (2006) Linear systems. Birkhauser, BostonGoogle Scholar
2. DeCarlo RA (1989) Linear systems. Prentice-Hall, Englewood CliffsGoogle Scholar
3. Kailath T (1980) Linear systems. Prentice-Hall, Englewood CliffsGoogle Scholar
4. Rugh WJ (1996) Linear systems theory, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
5. Sontag ED (1990) Mathematical control theory: deterministic finite dimensional systems. Texts in applied mathematics, vol 6. Springer, New YorkGoogle Scholar

## Copyright information

© Springer-Verlag London 2014

## Authors and Affiliations

1. 1.Department of Electrical EngineeringUniversity of Notre DameNotre DameUSA